presenting degenerate ringel–hall algebras of cyclic quivers

Journal of Pure and Applied Algebra 214 (2010) 1787–1799

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Journal of Pure and Applied Algebra

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Presenting degenerate Ringel–Hall algebras of cyclic quiversBangming Deng a,∗, Jie Du b, Alexandre Mah ba School of Mathematical Sciences, Beijing Normal University, Beijing 100875, Chinab School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia

a r t i c l e i n f o

Article history:Received 12 August 2009Available online 15 January 2010Communicated by E.M. Friedlander

MSC: 17B37; 16G20

a b s t r a c t

Using the generators labelled by simple and sincere semisimple modules for theRingel–Hall algebraHq(n) of a cyclic quiver∆(n), we give a presentation for the degeneratealgebra H0(n). This is achieved by establishing a presentation for the generic extensionmonoid algebra of∆(n). As an application, we show that both the degenerate Ringel–Hallalgebra and the degenerate quantum affine sln admit multiplicative bases.

© 2009 Elsevier B.V. All rights reserved.

Following the remarkable realization [13,14] of the ±-part of quantum enveloping algebras of finite type in terms ofHall algebras, Ringel [15] introduced the generic Hall algebra Hq(n) associated with a cyclic quiver ∆(n) (n > 2) andshowed that its composition subalgebra is isomorphic to the±-part of the quantum enveloping algebraUv(sln). Though thisapproach to quantum enveloping algebras has been extended both geometrically and algebraically to all types associatedwith symmetrizable Cartan matrices, these Hall algebras, known as Ringel–Hall algebras, of finite or cyclic type are theonly ones defined by the existence of Hall polynomials. Thus, their degenerate version is well defined by specializing q to0. The structure of the degenerate Ringel–Hall algebras of finite type has been studied in [11]. This paper investigates thedegenerate Ringel–Hall algebra H0(n) associated with∆(n).The structure of the Ringel–Hall algebra Hq(n) in terms of its composition subalgebra together with infinitely many

central elements has been described in [16]. However, the algebraic description of these central elements is rathercomplicated; see [8]. On the other hand, as seen in [5], Hq(n) has natural generators labelled by simple and sinceresemisimple modules. It would be interesting to describe the relations between these generators via which a presentationcan be derived. As a first attempt, we provide in this paper such a presentation of the degenerate Ringel–Hall algebra H0(n).Our approach is to use generic extensions of representations of a quiverQ studied in [1]. In general, the generic extension

of any two representations may not exist. However, it was shown in [11] that, if Q is a Dynkin quiver, then the genericextension of any two representations always exists. Moreover, taking generic extensions defines a monoid structure on theset of isoclasses (=isomorphism classes) of representations of Q . It turns out that this monoid structure provides a powerfultool in constructing bases of the corresponding quantized enveloping algebra in terms of Ringel–Hall algebra of Q ; see forexample [11,12,3,6]. By modifying [11], it was shown in [2] that the generic extension of any two nilpotent representationsof a cyclic quiver ∆(n) also exists. Thus, a monoid structure on the setM(n) of isoclasses of nilpotent representations of∆(n) is defined. As a generalization of [15], a systematic construction of monomial bases for quantum enveloping algebraUv(sln)was presented in [2]. Moreover, by establishing a relation between monomial bases and PBW bases, an elementaryalgebraic construction of canonical bases for both U+v (sln) and Hq(n)was obtained in [5].For the purpose of this paper, we study the monoid algebra ZM(n). More precisely, we will describe the generating

relations for ZM(n) in terms of a minimal set of generators ofM(n) and show that ZM(n) is in fact isomorphic to H0(n).In particular, both the degenerate Ringel–Hall algebra H0(n) of∆(n) and its degenerate composition subalgebra (called the

∗ Corresponding author.E-mail addresses: [email protected] (B. Deng), [email protected] (J. Du), [email protected] (A. Mah).URL: http://web.maths.unsw.edu.au/∼jied (J. Du).

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1788 B. Deng et al. / Journal of Pure and Applied Algebra 214 (2010) 1787–1799

degenerate quantum affine sln) admit multiplicative bases. We remark that the isomorphism between ZM(n) and H0(n)without involving generators and relations has been obtained independently in [17] by a direct calculation of certain Hallpolynomials (see Remark 3.4).The paper is organized as follows. In Section 1 we introduce the degenerate Ringel–Hall algebra H0(n), derive a number

of relations between the generators, and state the main theorem (Theorem 1.6). In Section 2 we recall the notion of genericextensions for nilpotent representations of ∆(n), and define the generic extension monoidM(n). We then state a similarpresentation for the generic extension monoid algebra ZM(n) in Theorem 2.4 and show that Theorem 1.6 follows fromTheorem 2.4. Before proving Theorem 2.4 in the last section, we give some applications of the main theorem in Section 3dealing with multiplicative bases for H0(n) and the degenerate quantum affine sln.

1. The degenerate Ringel–Hall algebraH0(n)

Let ∆ = ∆(n) (n > 2) be the cyclic quiver with vertex set I := Z/nZ = 1, 2, . . . , n and arrow set i −→ i + 1 | 1 ≤i ≤ n. Let k be a field. For a representation M = (Vi, fi)i of ∆ over k (equivalently, a module over the path algebra k∆ of∆), let dimM =

∑ni=1 dim Vi and dimM = (dim V1, . . . , dim Vn) ∈ N

n denote the dimension and dimension vector of M ,respectively, and let [M] denote the isoclass ofM .A representation M = (Vi, fi)i of ∆ is called nilpotent if the composition fn · · · f2f1 : V1 → V1 is nilpotent. Let Rep 0k∆

denote the category of finite dimensional nilpotent representations of∆ over k, and let Si = (Si)k, i ∈ I , be the simple objectsin Rep 0k∆. It is well known that for each i ∈ I and each integer l > 1, there is a unique (up to isomorphism) indecomposableobject Sidlek in Rep 0k∆with top (Si)k and dimension l. See Remark 2.2 for a dual version.Following [9], letΠ denote the set of formal finite sums (called multisegments)

π =∑i∈I,l>1

πi,l[i; l),

where πi,l ∈ N. Each multisegment π =∑i,l πi,l[i; l) ∈ Π defines a representation in Rep

0k∆1

M(π) = Mk(π) =⊕i∈I, l>1

πi,lSidlek.

In this waywe obtain a bijection betweenΠ and the setM of isoclasses of representations inRep 0k∆. Note that this bijectionis independent of the field k.For each d ∈ Nn, setΠd = π ∈ Π | dimM(π) = d.

ThenΠ =⋃

d∈Nn Πd . By identifying Nn with a subset ofΠ via the map

Nn −→ Π, a = (ai) 7−→ πa :=

n∑i=1

ai[i; 1), (1.0.1)

the modules Sa := M(πa) =⊕ni=1 aiSi, for a ∈ N

n, form a complete set of semisimple modules in Rep 0k∆. The semisimplemodule Sa is called sincere if a = (ai) ∈ Nn is a sincere vector, i.e., supp a := i ∈ I | ai 6= 0 = I . Let

J = all sincere vectors in Nn and I = I ∪ J.If k is a finite field, then for given modulesM,N1, . . . ,Nm in Rep 0k∆, the number of the filtrationsM = M0 ⊇ M1 ⊇ · · · ⊇ Mm−1 ⊇ Mm = 0

satisfyingMt−1/Mt ∼= Nt , for all 1 ≤ t ≤ m, is finite. Denote this number by FMN1,...,Nm . By [15,7], FMN1,...,Nm

is a polynomial.Moreprecisely, for π,µ1, . . . , µm inΠ , there is a polynomial ϕπµ1,...,µm(q) ∈ Z[q] (the polynomial ring over Z in indeterminate q)such that for any finite field k of qk elements,

ϕπµ1,...,µm(qk) = FMk(π)Mk(µ1),...,Mk(µm)

.

The (generic) Ringel–Hall algebra H = Hq(n) of ∆ = ∆(n), following [13], is the free module over Z[q] with basisuπ | π ∈ Π and multiplication given by

uµuν =∑π∈Π

ϕπµ,ν(q)uπ .

It is an Nn-graded algebra

H =⊕d∈Nn

Hd, (1.0.2)

where Hd is spanned by all uπ , π ∈ Πd . Also, by [5, Th. 5.2(i)], H is generated by ua, a ∈ I , where a identifies with πavia (1.0.1).

1 In [15,2], nilpotent representations of∆ are parametrized by using n-tuples of partitions.

B. Deng et al. / Journal of Pure and Applied Algebra 214 (2010) 1787–1799 1789

The notation uπ = u[M(π)] will be used when the relevant modules are involved in calculations. In particular, ua = u[Sa]for each a ∈ Nn. We also write ui = uei = u[Si] for each i ∈ I , where

ei := (. . . , 0, 1(i), 0, . . .)

is the ‘‘standard basis’’ element of Nn.Let Ω (resp.,Ω) denote the set of allwords in the alphabet I (resp., I). Given awordw = a1a2 · · · am in Ω , we can uniquely

expressw in the tight formw = bn11 bn22 · · · bntt , where nr = 1 if br ∈ J , and nr is the number of consecutive occurrences of br

if br ∈ I . Define

uw = ua1ua2 · · · uam and u(w) = un1b1un2b2 · · · untbt .

Since umei = umi /[[m]]

! (see, for example, [4, (10.2.1)]), where [[m]]! = [[1]][[2]] · · · [[m]] with [[i]] = qi−1q−1 for all m > 1, we

have

u(w) =1

t∏r=1[[nr ]]!

uw. (1.0.3)

By [5], there is a surjective map ℘ : Ω → Π (see (2.1.2) for the definition). For a word w = a1a2 · · · am in Ω with tightform w = bn11 bn22 · · · b

ntt , we denote by γ πw (q) the Hall polynomial ϕ

πµ1,...,µt

(q), where µr = nrbr for 1 6 r 6 t . The word wis called distinguished if γ ℘(w)w (q) = 1. For each π ∈ Π , fix a word wπ ∈ ℘−1(π). We call the set wπ | π ∈ Π a sectionof Ω over Π . Such a section is called distinguished if all wπ are distinguished words. By [5, Prop. 4.5], there always existdistinguished sections of Ω overΠ .Now, by specializing q to 0, we obtain the Z-algebra H0(n), called the (integral) degenerate Ringel–Hall algebra associated

with∆(n). In other words,

H0(n) = Hq(n)⊗Z[q] Z,

where Z is viewed as a Z[q]-module with the action of q being zero. By abuse of notation, we also write uπ = uπ ⊗ 1,uw = uw ⊗ 1, etc. Thus, uπ | π ∈ Π is a Z-basis of H0(n). The following result shows that the subset ua | a ∈ I of thisbasis generates H0(n).

Lemma 1.1. The Z-algebra H0(n) is generated by ua, a ∈ I . Moreover, umei = umi in H0(n) for all i ∈ I and m ∈ N.

Proof. By [5, Th. 5.2(ii)], for each distinguished section wπ π∈Π , the set u(wπ ) | π ∈ Π forms a basis for Hq(n). Since thevalue [[m]]0 of [[m]] at q = 0 is 1 for allm > 0, we have u(wπ ) = uwπ in H0(n). It follows that uwπ π∈Π is a Z-basis for H0(n).Hence, H0(n) is generated by ua, a ∈ I .

Consider the bijective map

τ : Nn −→ Nn, a = (a1, a2, . . . , an) 7−→ τa = (an, a1, . . . , an−1).

For each pair (a, b) of elements a = (ai), b = (bi) ∈ J , we define2

min(a, b) := c = (c1, . . . , cn) and (a, b) := (a+ b− τc, τc), (1.1.1)

where ci = minai, bi+1 for all i ∈ I .For a, b ∈ Nn, we write a 6 b if ai 6 bi for all i ∈ I , i.e., if b− a ∈ Nn.

Lemma 1.2. Maintain the notation introduced above. For a, b ∈ J , we have(1) (a, b) = (a, b) if and only if b = τc;(2) b 6 τ a;(3) for u, v ∈ J , (u, v) = (a, b) if and only if (u, v) = (c + x, τc + y) for all (x, y) ∈ Ξ(z), where

Ξ(z) = (x, y) | x, y ∈ Nn, x+ y = z, supp τx ∩ supp y = ∅

and z = a− c + b− τc .

Proof. Statement (1) is clear from the definition. To see (2), we observe

(τ a)i = (τa)i + (τb)i − (ττc)i = ai−1 + bi−1 − ci−2.

Since bi−1 − ci−2 > 0 and ai−1 > ci−1, it follows that (τ a)i > (b)i for all i, proving (2). Finally, the ‘‘if part’’ of (3) is clearfrom definition. Suppose now (u, v) = (a, b). Then min(u, v) = min(a, b), i.e., minai, bi+1 = minui, vi+1 for all i, andu+ v = a+ b. Let x = u− c and y = v− τc . It is straightforward to check that (x, y) ∈ Ξ(z), proving (3).

2 It would bemore appropriate to define (a, b) := (a+b−τc, τc). Since both component of (a, b)will be separately used later on, we put a hat on eachcomponent to write (a, b) for (a, b). The reader is warned that hatting on the first variable has a different meaning from hatting on the second variable.


We are ready to describe certain relations among the generators ua, a ∈ I of H0(n).

Lemma 1.3. For i, j ∈ I , a = (al), b = (bl) ∈ J , the following relations hold in H0(n):

(R1) uiua+ei+1 = ua+eiui+1;(R2) uiua = u

ai+1i uai−1i−1 · · · u

ai−n+2i−n+2ui+1, whenever ai+1 = 1;

(R3) uaui+1 = uiuai−1i−1 · · · u

ai−n+2i−n+2u

ai+1+1i+1 , whenever ai = 1;

(R4) uaub = uaub, whenever b τa;

and, in addition, for n > 3,

(R5) uiuj = ujui, whenever j 6≡ i± 1mod n;(R6) u2i ui+1 = uiui+1ui;(R7) uiu2i+1 = ui+1uiui+1.

Proof. Relations (R5)–(R7) follow from [15, 8.7] by setting q = 0.Relations (R1)–(R3) follow from the following formulas in Hq(n): for i ∈ I, b, c ∈ J ,

uiub = u[Sid2e⊕Sb−ei+1 ] + [[bi + 1]]ub+ei ,ucui+1 = u[Sid2e⊕Sc−ei ] + [[ci+1 + 1]]uc+ei+1 .

(1.3.1)

Hence, taking b = a+ ei+1, c = a+ ei and q = 0 gives

uiua+ei+1 = u[Sid2e⊕Sa] + ua+ei+ei+1 = ua+eiui+1

in H0(n), proving (R1).We now prove (R2). Relation (R3) can be proved similarly. For a ∈ J and i ∈ I , put

ua,i := uaieiuai−1ei−1 · · · uai−n+1ei−n+1 ∈ Hq(n).

Then ua,i = uaii uai−1i−1 · · · u

ai−n+1i−n+1 in H0(n). Suppose a ∈ J with ai+1 = 1. Then in Hq(n),

uiua,i = uiuaieiuai−1ei−1 · · · uai−n+1ei−n+1= [[ai + 1]]u[Sid2e⊕Sa−ei+1 ] + [[ai + 1]]ua+ei .

Therefore, (1.3.1) implies uiua = u[Sid2e⊕Sa−ei+1 ] + ua+ei = uiua,i in H0(n).It remains to prove (R4). For a, b ∈ J , let c = min(a, b) be defined as in (1.1.1). For each d = (di) ∈ Nn satisfying d 6 c ,

define κd ∈ Π by

M(κd) =⊕i∈I

(diSid2e ⊕ (ai + bi − di−1 − di)Si

).

It is clear that M(κd) is an extension of Sa by Sb, and each extension of Sa by Sb is isomorphic to some M(κd) with d 6 c .Hence, in Hq(n),

uaub =∑

d∈Nn, d6c

ϕκda,b(q)uκd .

An easy calculation shows that

ϕκda,b(q) =

∏i∈I

[[ai + bi − di−1 − di

bi − di−1

]].

Thus, in H0(n),

uaub =∑

d∈Nn,d6c

uκd .

In other words, the product uaub is completely determined by min(a, b). Since min(a, b) = min(a, b), it follows thatuaub = uaub in H0(n), as required.

Remark 1.4. Relation (R1) combines the case in (R2) when ai+1 > 2 and the case in (R3) when ai > 2. This relation hasthe form uiuc = udui+1. Hence, we call it the shifted commutative relation between simple and sincere semisimple modules.Relations (R2) and (R3) are the so-called cyclic-linear relations because it turns a semisimple representation of a cyclic quiverto a representation of a linear quiver. Relation (R4) is used to turn a monomial associated with a word in the alphabet I intomonomial associated with a standard word (see Definition 4.2(3)). So we call (R4) the standard word relation. Obviously,relations (R5)–(R7) are the degenerate quantum Serre relations.


Corollary 1.5. Maintain the notation above. For any a, b ∈ J with c = min(a, b),

uaub =∑

d∈Nn,d6c

∏i∈I

[[ai + bi − di−1 − di

bi − di−1

]]uκd

in Hq(n).

The main result of the paper is the following theorem.

Theorem 1.6. The generators ua, a ∈ I , and relations (R1)–(R7) form a presentation of H0(n).

The proof of the theorem is reduced to proving a similar presentation for the generic extension monoid algebra for∆(n)which is discussed in the next section.

2. Reduction to the generic extension monoid algebras

LetM =M(n) be the set of all isoclasses of representations in Rep 0k∆. Given two objectsM,N in Rep0k∆, there exists a

unique (up to isomorphism) extension G ofM by N with minimal dim End k∆(G) (see [1,11,2]). The extension G is called thegeneric extension ofM by N and is denoted byM ∗ N . Thus, if we define [M] ∗ [N] = [M ∗ N], then it is known from [2] that∗ is associative and (M, ∗) is a monoid with identity [0].For π, π ′ ∈ Π , define π ∗ π ′ ∈ Π by

M(π ∗ π ′) ∼= M(π) ∗M(π ′).

The multisegment π ∗ π ′ ∈ Π can be recursively computed as follows.First, by identifying i ∈ I with [i; 1) ∈ Π , [2, Prop. 3.7] implies

Si ∗M(π) ∼= M(i ∗ π),

where i ∗ π = π − [i+ 1; l0)+ [i; l0 + 1)with l0 being the maximal index such that πi+1,l0 6= 0.Second, if we write π = π (1) + π (2) + · · · + π (n), where, for each i ∈ I , π (i) =

∑l>1 πi,l[i; l), then we have the following

result.

Lemma 2.1 ([5, Lemma 3.1]). For a ∈ Nn and π ∈ Π , M(a ∗ π) ∼= Sa ∗M(π), where

a ∗ π =∑i∈I

i ∗ i ∗ · · · ∗ i︸︷︷︸ai

∗π (i+1).

Finally, following this lemma, for each M ∈ Rep 0k∆ with Loewy length l =: Ll(M), the radical and socle filtrations (see[5, 1.1]) give rise to

M ∼= (M/radM) ∗ (radM/rad 2M) ∗ · · · ∗ (rad l−2M/rad l−1M) ∗ (rad l−1M)M ∼= (M/soc l−1M) ∗ (soc l−1M/soc l−2M) ∗ · · · ∗ (soc 2M/socM) ∗ (socM).

(2.1.1)

Thus, there exist elements a1, . . . , al ∈ Nn such that

π ∗ π ′ = a1 ∗ · · · ∗ al ∗ π ′.

Consequently, we obtain a surjective map (cf. (1.0.1))

℘ : Ω −→ Π, a1 · · · am 7−→ a1 ∗ · · · ∗ am. (2.1.2)

We will also view ℘ as a map Ω → M taking w 7→ [M(℘(w))]. This is called the generic extension map which has niceapplication to the description of monomial bases for Hq(n); see [5, Th. 5.2].

Remark 2.2. There is a dual version of the above formulas. Let Siblck be the indecomposable module in Rep 0k∆ with socle(Si)k and dimension l and, for π =

∑i,l πi,l[i; l) ∈ Π , let

W (π) =⊕i∈I, l>1

πi,lSiblck.

If we define π ? π ′ byW (π ? π ′) = W (π) ∗W (π ′), then for i ∈ I and a ∈ Zn,

π ? i = π − [i− 1; l0)+ [i; l0 + 1),

where l0 is the maximal index such that πi−1,l0 6= 0, and

π ? a =∑i∈I

π (i−1) ? i ? i ? · · · ? i︸︷︷︸ai

.

Since every non-sincere semisimple module can be generated by simple modules, the above discussion shows that thegeneric extension monoidM(n) is generated by [Sa], a ∈ I (see [5, Prop. 3.3]). This gives the first assertion of the followingresult. Moreover, Lemma 2.1 can also be used to check that relations similar to Lemma 1.3(R1)–(R7) hold in ZM(n).


For a ∈ J and i ∈ I , put

[Sa,i] = [Saiei ] ∗ [Sai−1ei−1 ] ∗ · · · ∗ [Sai−n+1ei−n+1 ] ∈ ZM(n).

Lemma 2.3. If ZM(n) is the monoid algebra ofM(n) over Z, then ZM(n) is generated by [Sa], a ∈ I . Moreover, the followingrelations hold in H0(n):

(R1′) [Si] ∗ [Sa+ei+1 ] = [Sa+ei ] ∗ [Si+1];(R2′) [Si] ∗ [Sa] = [Si] ∗ [Sa,i], whenever ai+1 = 1;(R3′) [Sa] ∗ [Si+1] = [Sa,i] ∗ [Si+1], whenever ai = 1;(R4′) [Sa] ∗ [Sb] = [Sa] ∗ [Sb], whenever b τa;

and, in addition, for n ≥ 3,

(R5′) [Si] ∗ [Sj] = [Sj] ∗ [Si], whenever j 6≡ i± 1mod n;(R6′) [Si] ∗ [Si] ∗ [Si+1] = [Si] ∗ [Si+1] ∗ [Si];(R7′) [Si] ∗ [Si+1] ∗ [Si+1] = [Si+1] ∗ [Si] ∗ [Si+1],

for all i, j ∈ I, a = (al), b = (bl) ∈ J .

In fact, these relations provide a presentation of ZM(n).

Theorem 2.4. The monoid algebra ZM(n) has a presentation with generators [Sa], a ∈ I and relations (R1′)–(R7′).

Like the Ringel–Hall algebra, there is a natural grading on ZM(n) in terms of dimension vectors:

ZM(n) =⊕d∈Nn

ZM(n)d,

where ZM(n)d is spanned by all [M(π)], π ∈ Πd .

Corollary 2.5. For each n > 2, there is a graded Z-algebra isomorphism

φ : ZM(n) −→ H0(n), [Sa] 7−→ ua, a ∈ I.

Proof. By Theorem 2.4 and Lemmas 1.1 and 1.3, there is a surjective Z-algebra homomorphism

φ : ZM(n) −→ H0(n), [Sa] 7−→ ua, a ∈ I.

Choose a distinguished section wπ π∈Π . Then ℘(wπ ) = π and the set uwπ | π ∈ Π forms a basis for H0(n). Since[M(π)] | π ∈ Π is a basis for ZM(n), and φ([M(π)]) = uwπ , it follows that φ is an isomorphism.

Thus, we have reduced the proof of Theorem 1.6 to proving Theorem 2.4 which will be given in Section 4.

3. Applications: Multiplicative bases and degenerate quantum groups

Given two representations M,N ∈ Rep 0k∆, we say that M degenerates to N (or N is a degeneration of M), written asM 6deg N , if dimM = dimN and

dimHomk∆(X,M) > dimHomk∆(X,N)

for all X in Rep 0k∆. See [1,18,4].Since the order relation is independent of the field k, we may turnΠ into a poset with the partial order 6=6deg defined

by setting

µ 6 λ⇐⇒ M(µ) 6deg M(λ).

Applying Corollary 2.5 gives a multiplicative basis of H0(n). Namely, for each π ∈ Π , define θπ = φ([M(π)]). Thenθπ | π ∈ Π is a basis of H0(n). It is clear that θπθπ ′ = θπ∗π ′ . In other words, θπ | π ∈ Π is a multiplicative basis ofH0(n). For each π ∈ Π , write

θπ =∑λ∈Π

aπ,λuλ. (3.0.1)

We now determine the coefficients aπ,λ.

Proposition 3.1. For π, λ ∈ Π , aπ,λ 6= 0 implies λ 6 π , and, moreover, aπ,π = 1.

Proof. By [5, Prop. 4.5], we fix a distinguished wordw = a1a2 · · · am ∈ ℘−1(π). Then, by [5, Lemma 5.1],

uw = ua1ua2 · · · uam =t∏r=1

[[nr ]]!∑

λ6℘(w)

γ λw(q)uλ.


Thus,

θπ = φ([M(π)]) = φ([Sa1 ] ∗ [Sa2 ] ∗ · · · ∗ [Sam ])= φ([Sa1 ])φ([Sa2 ]) · · ·φ([Sam ])

= ua1ua2 · · · uam = uπ +∑λ<π

γ λw(0)uλ.

Hence, aπ,π = 1, and aπ,λ = 0 unless λ 6 π . Moreover, aπ,λ = γ λw(0) for λ 6 π .

Given a wordw = a1a2 · · · am ∈ Ω and a moduleM ∈ Rep 0k∆, a filtration

M = M0 ⊇ M1 ⊇ · · · ⊇ Mm−1 ⊇ Mm = 0

ofM is said to be of typew ifMt−1/Mt ∼= Sat for all 1 ≤ t ≤ m. The following proposition was proved in [17, Th. 2.7].

Proposition 3.2. Letw = a1a2 · · · am ∈ Ω and λ ∈ Π . If M(λ) has a filtration of typew, then ϕλa1,...,am(0) = 1.

Since γ λw(0) = ϕλa1,...,am(0) for any word w = a1a2 · · · am ∈ Ω , the above two propositions have the following directconsequence.

Corollary 3.3. For each π ∈ Π , we have aπ,λ = 1 whenever λ 6 π , and aπ,λ = 0 otherwise. In other words, (3.0.1) can berefined as θπ =

∑λ6π uλ.

Remark 3.4. By a direct calculation of certain Hall polynomials,Wolf established in [17] the isomorphism given in Corollary2.5 through the map

ZM(n) −→ H0(n), [M(λ)] 7−→ θλ.

The next application is closely related to the Lusztig form of quantum affine sln. Denote by C = Cq(n) theZ[q]-subalgebraof H generated by umei , i ∈ I , m > 1. This is called the (generic) composition algebra of ∆ = ∆(n). (Its twisted version overZ[q

12 , q−

12 ] is known as the Lusztig form; see [5, Section 6].) It is easy to see that C is a proper subalgebra of H. Moreover,

the Nn-grading on Hq(n) induces a grading on C:

C =⊕d∈Nn

(C ∩ Hd). (3.4.1)

By specializing q to 0, the resulting algebra identifies with the subalgebra C0(n) of H0(n) generated by umei , i ∈ I , m > 1.This is called the (integral) degenerate quantum affine sln. It also inherits the Nn-grading.

Remark 3.5. The subalgebra C′ of Hq(n) generated by ui (i ∈ I) is a proper subalgebra of Cq(n) since form > 1 and i ∈ I ,

umei =1[[m]]!

umi 6∈ C′.

However, the equality umei = umi holds in H0(n). Thus, C0(n) coincides with the subalgebra of H0(n) generated by ui, i ∈ I .

The monoidM(n) admits a submonoidMc = Mc(n) generated by [Si], i ∈ I , called the composition monoid of ∆. It isshown in [2] thatMc consists of the isoclasses [M(π)]with π ∈ Πa, whereΠa denotes the set of aperiodic multisegments,that is, those π ∈ Π satisfying for each l > 1, there is some i ∈ I such that πi,l = 0. The π ∈ Π\Πa are called periodic. Therestriction of the map ℘ given in (2.1.2) induces a surjection ℘ : Ω Πa (see [2, Th. 4.1]).Also, the monoid algebra ZMc(n) is naturally Nn-graded with

ZMc(n)d =⊕d∈Nn

ZMc(n) ∩ ZM(n)d .

Proposition 3.6. For each n > 2, the isomorphism φ : ZM(n) −→ H0(n) induces a graded Z-algebra isomorphism

φc : ZMc(n) −→ C0(n), [Si] 7−→ ui, i ∈ I.

In particular, θπ | π ∈ Πa is a multiplicative basis of C0(n).

In [5, Section 7], a PBW-like basis for the quantumaffine slnwas constructed in terms of the (twisted) composition algebraof∆(n). This basis plays an important role in the elementary algebraic construction of the canonical basis. We now imitatethis construction to obtain a PBW-like Z-basis of Cq(n)which degenerates a basis of C0(n).By [5, Prop. 4.3], for each π ∈ Πa, there exists a distinguished wordwπ ∈ Ω ∩℘−1(π). Let j

n11 jn22 · · · j

ntt be the tight form

ofwπ . By (1.0.3) and [5, Lemma 5.1],

u(wπ ) = un1ej1 un2ej2 · · · unt ejt = uπ +∑λ<π

γ λwπ (q)uλ.


Now take a distinguished section D = wπ | π ∈ Πa. Let d ∈ Nn. If π ∈ Πad is minimal with respect to 6=6deg, put

eπ = u(wπ ) ∈ Cq(n)d . In general, for π ∈ Πad , assuming that all eλ with λ < π have been already defined, we define

eπ = u(wπ ) −∑

λ∈Πad ,λ<π

γ λwπ (q)eλ ∈ Cq(n)d .

Thus, we have

eπ = uπ +∑

λ∈Πd\Πad ,λ<π

ζ πλ (q)uλ,

where ζ πλ (q) ∈ Z[q].An argument similar to that in [5, Th. 7.5] gives the following result.

Proposition 3.7. LetD = wπ | π ∈ Πa be a distinguished section ofΩ . Then both u(wπ ) | π ∈ Πa and eπ | π ∈ Πa arebases of Cq(n) over Z[q]. In particular, Cq(n) is a free Z[q]-module.

Also, applying a similar argument as in [5, Cor. 8.3] shows that the basis eπ | π ∈ Πa is independent of the choice ofthe distinguished sectionD .For π ∈ Πa, set

eπ := eπ |q=0 = uπ +∑

λ∈Πd\Πad ,λ<π

ζ πλ (0)uλ ∈ C0(n).

The above proposition implies that eπ | π ∈ Πa is a Z-basis of C0(n). By the proof of Proposition 3.1, the elements eπ canbe also obtained from the θπ recursively, namely, for each π ∈ Πad ,

eπ = θπ −∑

λ∈Πd\Πad , λ<π

eλ.

4. Proof of Theorem 2.4

We now prove Theorem 2.4. Let A = A (n) be the free Z-algebra with generators sa, a ∈ I . Clearly, A has a monomialbasis sww∈Ω , where sw := sa1 · · · sam with ai ∈ I . For a ∈ N

n and i ∈ I , define elements

sa,i := saii sai−1i−1 · · · s

ai−n+1i−n+1 .

Consider the ideal I of A (n) generated by the following elements (cf. [10]): for i, j ∈ I, a = (al), b = (bl) ∈ J,(E1) sisa+ei+1 − sa+eisi+1;(E2) sisa − sisa,i, whenever ai+1 = 1;(E3) sasi+1 − sa,isi+1, whenever ai = 1;(E4) sasb − sasb, whenever b τa;

and, in addition, for n ≥ 3,

(E5) sisj − sjsi, whenever j 6≡ i± 1 mod n;(E6) s2i si+1 − sisi+1si;(E7) sis2i+1 − si+1sisi+1.

Thus, the surjective algebra homomorphism Ψ : A (n) → ZM(n), sa 7→ [Sa] induces, by Lemma 2.3, a surjective algebrahomomorphism Ψ : A (n)/I → ZM(n).Now, to complete the proof of Theorem2.4, it suffices to show thatΨ is an isomorphism. The rest of this section is devoted

to proving this fact.Given two elements x, y ∈ A (n), we say that x and y are equivalent,write x ∼ y, if x−y ∈ I . This is clearly an equivalence

relation on A (n). Also,

x ∼ y =⇒ z1xz2 ∼ z1yz2 for all z1, z2 ∈ A (n). (4.0.1)

In particular, (E5) implies that if i, j ∈ I are not adjacent in∆(n), then sisj ∼ sjsi.

Lemma 4.1. Let a = (al) ∈ Nn be a non-sincere vector and assume ai = 0. ThenΨ (sa,i) = [Sa], and forw ∈ Ω , sw ∼ sa,i if andonly ifw ∈ ℘−1([Sa]). In particular, if aj = 0 with j 6= i, then sa,i ∼ sa,j.

Proof. The first assertion is clear since

Ψ (sa,i) = [Si−1]ai−1 ∗ [Si−2]ai−2 ∗ · · · ∗ [Si−n+1]ai−n+1 =[i−n+1⊕j=i−1

ajSj

]= [Sa].

If a 6= 0 and ai = aj = 0 for i 6= j, then n ≥ 3. The last assertion follows from (E5).


We introduce, for a non-sincere a, the notation sa which leads to a number of generalizations.

Definition 4.2. (1) By definition, for each non-sincere vector a ∈ Nn, let sa denote a monomial representative of the formsw for somew ∈ ℘−1([Sa]). (If a = 0, set s0 = 1, the identity element.)

(2) For a word w = a1 . . . am with ai ∈ Nn\0, let sw := sa1 · · · sam . This is a well-defined element in A (n). Though swdepends on the selection of sai for those non-sincere ai, we have by (4.0.1) that changes of selection result in equivalentelements.

(3) Let Ω be the set of all words in the alphabet Nn\0. A word w = a1 · · · am in Ω is called standard if at+1 6 τat for all1 6 t < m.

(4) Forw = a1 . . . am ∈ Ω , letM(w) = Sa1 ∗ · · · ∗ Sam . Define the map

℘ : Ω −→M(n), w 7−→ [M(w)].

Clearly, the restriction of ℘ to Ω is the map defined in (2.1.2).

In the rest of the proof, we aim to prove that

(i) there is a one-to-one correspondence betweenM(n) and the set Ωstd of standard words in Ω;(ii) for every wordw ∈ Ω , there is a standard wordw′ ∈ Ω such that sw ∼ sw′ .

The following result establishes (i).

Theorem 4.3. The restriction of ℘ to Ωstd induces a bijection

℘std : Ωstd −→M(n), w 7−→ [M(w)].

In other words, every fibre ℘−1([M]), [M] ∈M(n), contains a unique standard word.

Proof. We first prove that ℘std is surjective. Let M ∈ Rep 0k∆ have Loewy length l. The first formula in (2.1.1) defines aword wM := a1 . . . al, where, for each 1 6 t 6 l, at ∈ Nn is defined by rad t−1M/rad tM ∼= Sat . We claim that wM is astandard word, giving the required surjectivity. Indeed, by definition, we need to show that at+1 6 τat for each 1 6 t < l.Suppose M = M(π) for some π =

∑i∈I,r>1 πi,r [i; r) ∈ Π . For each i ∈ I , let π

(i)=∑r>1 πi,r [i; r). Then for 1 6 t < l,

Sat ∼=⊕ni=1 ct,iSi+t−1, where ct,i =

∑r>t πi,r . Putting c t = (ct,1, . . . , ct,n), it is clear that c t+1 6 c t for 1 6 t < l. Also,

a1 = c1, a2 = (c2,n, c2,1, . . . , c2,n−1) = τc2 and, in general, at+1 = τ tc t+1. Thus,

at+1 = τ tc t+1 6 τ tc t = τ(τ t−1c t) = τat ,

i.e.,wM = a1 . . . al is standard.It remains to prove that ℘std is injective. Suppose w = a1 . . . al is a standard word and M = M(w). It suffices to prove

that w = wM . This is clear if l = 1. Assume now l > 1. Since M = Sa1 ∗ M(a2 . . . al) and a2 6 τa1, it follows thatRad(M) = M(a2 . . . al). Thus, by induction,w = wM .

We need some preparatory results for establishing (ii).

Lemma 4.4. If a = (ai) ∈ Nn with ai = 0 for some i ∈ I , then

sasi+1 ∼ sa+ei+1 and si−1sa ∼ sa+ei−1 .

Proof. By (4.0.1) and Lemma 4.1, we can assume sb = sb,i for b ∈ a, a+ ei+1, a+ ei−1. Then the two equivalences becomeequalities.

The following relation generalizes (E1) to an arbitrary a ∈ Nn.

Lemma 4.5. For a ∈ Nn and i ∈ I , we have

sa+eisi+1 ∼ sisa+ei+1 .

Proof. There is nothing to prove if a = 0. Assume a 6= 0 and that a is non-sincere since the sincere case follows from (E1).Then at least one of the a+ ei and a+ ei+1 is non-sincere. Thus, we consider the following three cases.Case 1. a 6∈ J , but a+ ei+1 ∈ J . Then ai+1 = 0 and ai′ 6= 0 for all i′ 6= i+ 1. Thus, a+ ei 6∈ J . By (4.0.1) and Lemma 4.1, we

can take sa+ei = sa+ei,i+1, that is,

sa+ei = sa+ei,i+1 = sai+1i sai−1i−1 · · · s

ai−n+2i−n+2 = sis

aii sai−1i−1 · · · s

ai−n+2i−n+2 .

This implies that

sa+eisi+1 = sisaii sai−1i−1 · · · s

ai−n+2i−n+2 si−n+1 = sisa+ei+1,i

(noting i− n+ 1 = i+ 1 in I). Applying (E2) gives that sa+eisi+1 ∼ sisa+ei+1 .Case 2. a 6∈ J , but a + ei ∈ J . Then ai = 0 and ai′ 6= 0 for all i′ 6= i. Thus, a + ei+1 6∈ J . By taking sa+ei+1 = sa+ei+1,i, we

obtain

sa+ei+1 = sa+ei+1,i = sai−1i−1 s

ai−2i−2 · · · s

ai−n+1+1i−n+1 = sai−1i−1 s

ai−2i−2 · · · s

ai−n+1i−n+1 si+1.


Thensisa+ei+1 = sis

ai−1i−1 s

ai−2i−2 · · · s

ai−n+1i−n+1 si+1 = sa+ei,isi+1.

Hence, by (E3), sa+eisi+1 ∼ sisa+ei+1 .Case 3. a+ ei 6∈ J and a+ ei+1 6∈ J . Then n > 3, since n = 2 implies a = 0.If aiai+1 6= 0, then there exists i1 6= i, i+ 1 such that ai1 = 0. The selection of sa+ei = sa+ei,i1 and sa+ei+1 = sa+ei+1,i1 gives

sa+eisi+1 = sai1−1i1−1· · · sai+2i+2 s

ai+1i+1 s

ai+1i sai−1i−1 · · · s

ai1−n+1i1−n+1

si+1

∼ sai1−1i1−1· · · sai+2i+2 s

ai+1i+1 s

ai+1i si+1s

ai−1i−1 · · · s

ai1−n+1i1−n+1

(using (E5))

∼ sai1−1i1−1· · · sai+2i+2 s

ai+1i+1 sisi+1s


ai1−n+1i1−n+1

(using (E6) as ai 6= 0)

∼ sai1−1i1−1· · · sai+2i+2 sis

ai+1+1i+1 saii s

ai−1i−1 · · · s

ai1−n+1i1−n+1

(using (E7) as ai+1 6= 0)

∼ sisai1−1i1−1· · · sai+2i+2 s

ai+1+1i+1 saii s

ai−1i−1 · · · s

ai1−n+1i1−n+1

(using (E5))= sisa+ei+1 .

If aiai+1 = 0, there are three subcases to consider.(3a) If ai = ai+1 = 0, then the selection of sa+ei = sa+ei,i+1 and sa+ei+1 = sa+ei+1,i gives

sa+eisi+1 = sisai−1i−1 · · · s

ai2+1i2+1sai2−1i2−1· · · sai−n+2i−n+2 si+1 = sisa+ei+1 .

(3b) If ai 6= 0 and ai+1 = 0, then the selection of sa+ei = sa+ei,i+1 and sa+ei+1 = sa+ei+1,i2 , where i2 6= i, i + 1 and ai2 = 0,gives

sa+eisi+1 = sai+1i sai−1i−1 · · · s

ai2+1i2+1sai2−1i2−1· · · sai−n+2i−n+2 si+1

∼ sai+1i sai2−1i2−1· · · sai−n+2i−n+2 si+1s

ai−1i−1 · · · s

ai2+1i2+1

(using (E5))

∼ sai2−1i2−1· · · sai−n+2i−n+2 s

ai+1i si+1s

ai−1i−1 · · · s

ai2+1i2+1

(using (E5))

∼ sai2−1i2−1· · · sai−n+2i−n+2 sisi+1s


ai2+1i2+1

(using (E6))

∼ sisai2−1i2−1· · · sai−n+2i−n+2 si+1s


ai2+1i2+1

(using (E5))

= sisa+ei+1 (since si−n+2 = si+2 and si2+1 = si2−n+1).

(3c) If ai+1 6= 0 and ai = 0, then the selection of sa+ei = sa+ei,i1 and sa+ei+1 = sa+ei+1,i, where i1 6= i, i+ 1 and ai1 = 0, gives

sa+eisi+1 = sai1−1i1−1· · · sai+2i+2 s

ai+1i+1 sis

ai−1i−1 · · · s

ai1−n+1i1−n+1

si+1

∼ sai1−1i1−1· · · sai+2i+2 s

ai+1i+1 sisi+1s

ai−1i−1 · · · s

ai1−n+1i1−n+1

(using (E5))

∼ sai1−1i1−1· · · sai+2i+2 sis

ai+1+1i+1 sai−1i−1 · · · s

ai1−n+1i1−n+1

(using (E7))

∼ sisai1−1i1−1· · · sai+2i+2 s

ai+1+1i+1 sai−1i−1 · · · s

ai1−n+1i1−n+1

(using (E5))

∼ sisai−1i−1 · · · s

ai1−n+1i1−n+1

sai1−1i1−1· · · sai+2i+2 s

ai+1+1i+1 (using (E5))

= sisa+ei+1 (since si1−n+1 = si1+1, ai1 = 0, and si+1 = si−n+1).

The proof is completed. This lemma gives the following shifted commutative relations.

Corollary 4.6. For each a = (al) ∈ Nn, i ∈ I and 0 6 a 6 ai+1,sai sa ∼ sa−aei+1+aeis

ai+1.

We now introduce a partial ordering on Ω . Recall that the set Nn is a poset with the partial ordering: a 6 b⇐⇒ b− a ∈Nn. This induces a lexicographic partial ordering 4 on Ω by setting a1 · · · am 4 b1 · · · bt if either(1)m 6 t and al = bl for all 1 6 l 6 m, or(2) there exists 0 6 l 6 minm, t such that ar = br for 1 6 r < l and al < bl.

Lemma 4.7. Let a, b ∈ Nn. Suppose b 66 τa. Then there exist a′, b′ ∈ Nn such thatsasb ∼ sa′sb′ and ab ≺ a′b′.

Proof. Let a, b ∈ Nn satisfy b 66 τa. There are three cases to consider.If a, b ∈ J , i.e., a and b are sincere, then sasb ∼ sasb, by (E4), where (a, b) = (a + b − τc, τc) with c = min(a, b) as

defined in (1.1.1). Since b τa, it follows that b− τc 6= 0. Thus, a < a and hence, ab ≺ ab, proving the existence of a′ andb′ in this case.


Nowwe assume that b ∈ J and a 6∈ J . Suppose ai = 0 for some i ∈ I and choose sa = sa,i = sai−1i−1 s

ai−2i−2 · · · s

ai−n+1i−n+1 . If b > τa,

i.e., bj > aj−1 for all j ∈ I and at least one inequality is strict, then repeatedly applying Corollary 4.6 yields

sasb = sai−1i−1 s

ai−2i−2 · · · s

ai−n+1i−n+1 sb ∼ sb+a−τas

ai−1i sai−2i−1 · · · s

ai−n+1i−n+2 = sb+a−τasτa.

Thus, a′ = b + a − τa and b′ = τa satisfy the required properties. If b 6> τa, i.e., there is some j ∈ I such that bj < aj−1,then, since bi+1 > 0 = ai, there is d > 2 such that

bi+2 > ai+1, . . . , bi+d−1 > ai+d−2, but bi+d < ai+d−1.

Repeated applications of Corollary 4.6 gives

sasb = sai−1i−1 · · · s

ai+di+d s

ai+d−1i+d−1 s

ai+d−2i+d−2 · · · s

ai−n+1i−n+1 sb

∼ sai−1i−1 · · · sai+di+d s

ai+d−1−bi+di+d−1 scs

bi+di+d s

ai+d−2i+d−1 · · · s

ai−n+1i−n+2 ,

where

c = b− (ai+1ei+2 + · · · + ai+d−2ei+d−1 + bi+dei+d)+ (ai+1ei+1 + · · · + ai+d−2ei+d−2 + bi+dei+d−1).

Then ci+d = 0 and ci+d−1 = bi+d−1 − ai+d−2 + bi+d > 0 since b ∈ J . Lemma 4.4 impliessc ∼ si+d−1sc−ei+d−1 .

Hence, substituting and repeatedly applying Corollary 4.6 backwards yield

sasb ∼ sai−1i−1 · · · s

ai+di+d s

ai+d−1−bi+d+1i+d−1 sc−ei+d−1s

bi+di+d s

ai+d−2i+d−1 · · · s

ai−n+1i−n+2

∼ sai−1i−1 · · · sai+di+d s

ai+d−1+1i+d−1 sai+d−2i+d−2 · · · s

ai−n+1i−n+1 sb−ei+d−1

= sa+ei+d−1sb−ei+d−1 .

So a′ = a+ ei+d−1 and b′ = b− ei+d−1 are the required ones.Finally, we consider the case when b 6∈ J . Then by Lemma 4.1, we can take

sb = sb,i = sbi−1i−1 s

bi−2i−2 · · · s

bi−n+1i−n+1

for some i ∈ I with bi = 0. Since b 66 τa, there is d > 1 such thatbi−1 6 ai−2, . . . , bi−d+1 6 ai−d, but bi−d > ai−d−1.

Repeatedly applying Corollary 4.6 backwards gives

sasb = sasbi−1i−1 · · · s

bi−d+1i−d+1 s

bi−di−d s

bi−d−1i−d−1 · · · s

bi−n+1i−n+1

∼ sbi−1i−2 · · · sbi−d+1i−d sai−d−1i−d−1 scs

bi−d−ai−d−1i−d sbi−d−1i−d−1 · · · s

bi−n+1i−n+1 ,

where

c = a− (bi−1ei−2 + · · · + bi−d−1ei−d + ai−d−1ei−d−1)+ (bi−1ei−1 + · · · + bi−d+1ei−d+1 + ai−d−1ei−d).

Then ci−d−1 = 0. By Lemma 4.4, scsi−d ∼ sc+ei−d . Now substituting and applying Corollary 4.6 again yield

sasb ∼ sbi−1i−2 · · · s

bi−d+1i−d sai−d−1i−d−1 sc+ei−ds

bi−d−ai−d−1−1i−d sbi−d−1i−d−1 · · · s

bi−n+1i−n+1

∼ sa+ei−dsbi−1i−1 · · · s

bi−d+1i−d+1 s

bi−d−1i−d sbi−d−1i−d−1 · · · s

bi−n+1i−n+1

= sa+ei−dsb−ei−d .

So a′ = a+ ei−d and b′ = b− ei−d are the required ones in this last case. Theorem 4.8. For each word w ∈ Ω , there exists exactly one standard word w′ ∈ Ω such that sw ∼ sw′ . Moreover, each fibreof ℘ contains a unique maximal element which is the standard word of the fibre.

Proof. Since sw ∼ sw′ implies Ψ (sw) = Ψ (sw′), i.e., [M(w)] = [M(w′)], the uniqueness follows from Theorem 4.3. We nowprove the existence ofw′. Ifw itself is standard, we are done. Now suppose thatw is not standard.Wewritew = a1a2 · · · amwith a1, . . . , am ∈ Nn. Then there exists 1 6 t < m such that at+1 66 τat . By Lemma 4.7, there exist a′t , a

′

t+1 ∈ Nn such that

atat+1 ≺ a′ta′

t+1 and sat sat+1 ∼ sa′t sa′t+1 .

Put

w1 = a1 · · · at−1a′ta′

t+1at+2 · · · am.Then we have w ≺ w1 and sw ∼ sw1 . If w1 is standard, then we can take w

′= w1. Otherwise, we can repeat the above

process. So, we finally get a sequence of wordsw,w1, w2, . . . such that

w ≺ w1 ≺ w2 ≺ · · · and sw ∼ sw1 ∼ sw2 ∼ · · · .


In particular, dimM(wi) = dimM(w) for all i > 1. But there are only finitely many words v in Ω with dimM(v) =dimM(w). Hence, there must exist t > 1 such thatwt is standard. Lettingw′ = wt gives sw ∼ sw′ .

Now, the proof of Theorem 2.4 follows easily from Theorems 4.3 and 4.8.

Proof of Theorem 2.4. We need to show that the surjective algebra homomorphism Ψ : A (n)/I → ZM(n) is injective.By Theorems 4.3 and 4.8, the set

sw + I | w ∈ Ωstd

spans the Z-module A (n)/I and its image is a basis of ZM(n). Hence, Ψ is injective.

Remarks 4.9. (a) Consider the linear quiverL = Ln−1:

which is a full subquiver of∆(n). The category Rep kL of representations ofL over k can be viewed as a full subcategory ofRep 0k∆(n)which is closed under submodules, quotient modules and extensions. In other words, for π =

∑i,l πi,l[i, l) ∈ Π ,

M(π) is a representation ofL if and only if πn,l = 0 for all l > 1. Hence, the Ringel–Hall algebra Hq(L ) ofL identifies withthe Z[q]-subalgebra of Hq(n) of ∆(n) generated by umei for m > 1 and 1 6 i 6 n − 1. Also, its degenerate form H0(L ) canbe identified with the Z-subalgebra of H0(n) generated by ui, 1 6 i 6 n− 1.On the other hand, the generic extension monoid M(L ) of L identifies with the submonoid of M(n) (or Mc(n))

generated by [Si], 1 6 i 6 n− 1. Thus, the isomorphism φ in Corollary 2.5 induces a Z-algebra isomorphism

ZM(L ) −→ H0(L ), [Si] 7−→ ui, 1 6 i 6 n− 1.

By modifying the above proof of Theorem 2.4, we can obtain a presentation for ZM(L ), as well as H0(L ). To this aim,letB be the free Z-algebra with generators si, 1 6 i 6 n− 1, and letK be the ideal ofB generated by

(S1) sisj − sjsi, whenever |i− j| > 2;(S2) s2i si+1 − sisi+1si;(S3) sis2i+1 − si+1sisi+1,

for all 1 6 i, i+ 1, j 6 n− 1. Then, the natural mapB→ ZM(L ), si 7→ [Si] induces a surjective homomorphism

ψ : B/K −→ ZM(L ), si +K 7−→ [Si].

We now show that ψ is an isomorphism. Let

D = a = (ai) ∈ Nn | an = 0.

Furthermore, let D denote the set of all words in the alphabet in D\0which is a subset of Ω . Thus, ifw = a1a2 · · · am withat = (at,i) ∈ D is a standard word, i.e., at+1 6 τat for all 1 6 t < m, then at,i = 0 whenever i < t , and thus m 6 n − 1.As in the proof of Theorem 4.3, ifM is a representation of L with Loewy length l, then l 6 n− 1 and wM := a1a2 · · · al is astandard word in D, where at ∈ D is defined by rad t−1M/rad tM ∼= Sat for 1 6 t 6 l.For each a = (ai) ∈ D, we define

sa = san−1n−1 · · · s

a22 sa11 ∈ B,

and forw = a1a2 · · · am ∈ D, define

sw = sa1sa2 · · · sam ∈ B.

Clearly,B is spanned by all sw ,w ∈ D.We now define an equivalence relation on B by: x ∼′ y⇐⇒ x − y ∈ K . Then, it is easy to check that Lemmas 4.4, 4.5

and 4.7 and Corollary 4.6 continue to hold with respect to the equivalence relation∼′ for all a, b ∈ D, 1 6 i 6 n− 1. Hence,we obtain an analogue of Theorem 4.8. That is, for each word w ∈ D, there is a unique standard word w′ ∈ D such thatsw ∼′ sw′ . Consequently, ψ is an isomorphism.The isomorphism ψ gives a similar presentation for ZM(L ). In particular, the degenerate Ringel–Hall algebra H0(L ) is

generated by ui, 1 6 i 6 n− 1, with relations:

(1) uiuj = ujui for 1 6 i, j 6 n− 1 with |i− j| > 2;(2) u2i ui+1 − uiui+1ui for 1 6 i, i+ 1 6 n− 1;(3) uiu2i+1 − ui+1uiui+1 for 1 6 i, i+ 1 6 n− 1.

This provides an alternative proof of [11, Th. 4.2] for the linear quiver case.(b) The generating relations for the compositionmonoid algebra ZMc(n) are given in [10]. For example, it is shown there

that ZMc(2) is generated by [S1] and [S2]with relations:

(1) [S1]j ∗ [S2]j ∗ [S1]j+1 = [S1]j+1 ∗ [S2]j ∗ [S1]j, j > 1,(2) [S2]j ∗ [S1]j ∗ [S2]j+1 = [S2]j+1 ∗ [S2]j ∗ [S2]j, j > 1.

However, in general, it is very complicated to list the relations.


Acknowledgements

The authors would like to thank the referee for several helpful suggestions. The work is supported partially by theAustralianResearchCouncil, theNatural Science Foundation of China, and theDoctoral Programof ChineseHigher Education.The research was carried out while Deng was visiting the University of New South Wales. The hospitality and support ofUNSW are gratefully acknowledged.

References

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http://arxiv.org/0709.4348

presenting degenerate ringel–hall algebras of cyclic quivers

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